The cataclysmic-variable period gap is the observed deficit of accreting, ordinary hydrogen-rich cataclysmic variables with periods around two to three hours. The disrupted magnetic braking model interprets it as a detached phase after the donor contracts inside its Roche lobe.

For a Roche lobe-filling donor with local radius response $R_2\propto M_2^{\zeta }$, the Roche-lobe-filling period-density relation gives $P\propto M_2^{(3\zeta -1)/2}$. As mass is lost, the period falls for $\zeta >1/3$ and rises for $\zeta <1/3$, with a minimum at the transition. The reversal is called a period bounce. Slow evolution near the turning point produces an accumulation in the period distribution: for a steady flow of systems along one evolutionary branch, $N(P)\propto 1/|\dot{P}|$.

A classical Cassegrain reflector uses a concave parabolic primary and a convex hyperbolic secondary before the prime focus. The secondary returns the converging beam through a hole in the primary. It corrects on-axis spherical aberration but retains off-axis coma.

A Ritchey–Chrétien telescope has hyperbolic primary and secondary mirrors chosen to cancel third-order spherical aberration and coma. It retains astigmatism and field curvature; wide-field instruments may require additional correction.

A small weak charged current amplitude factor associated with $\sin {\theta }_C$. A doubly suppressed amplitude contains two such factors, giving a rate suppression of order ${\tan }^4{\theta }_C$ relative to a favoured process when other matrix elements are comparable.

An injective C-star homomorphism preserves norms. For unital maps, reduce to the commutative algebra generated by $1$ and $a^{\ast }a$. Its induced map on character spaces has dense, hence full, image; otherwise a nonzero continuous function vanishing on that image would lie in the kernel. The C-star identity then gives norm preservation for $a$.

A nonabelian simple group has no nonidentity conjugacy class of prime-power size. Coprime-degree irreducible characters vanish on such an element: the conjugacy-class sum makes the character-to-degree ratio an algebraic integer, and the Kronecker theorem on algebraic integers in the unit disk makes a nonzero ratio a root of unity, forcing a scalar in a faithful group representation. Column character orthogonality then makes $1/p$ an algebraic integer, impossible. This is the character-theoretic ingredient in Burnside's theorem.

A right-acting BRST differential obeys the right graded Leibniz rule. This convention permits $Q\varphi =[c,\varphi ]$ together with $Qc=-[c,c]/2$. Using a left graded Leibniz rule with those same two signs does not give a nilpotent operator; one of the signs must change.

Reflecting a Brownian motion $B$ after a stopping time $T$ by ${\hat{B}}_t=2B_T-B_t$ for $t>T$, and leaving it unchanged for $t\le T$, preserves its path probability distribution. For finite $T$, the Strong Markov property and symmetry of the future Brownian motion prove this. When $T=\infty$, the path is left unchanged.

A one-dimensional Brownian motion spends an infinite total time in every nonempty open interval almost surely. To prove this for $(-1,1)$ after first hitting zero, alternate exits from that interval with returns to zero. By recurrence of one-dimensional Brownian motion all these stopping times are finite, and the Strong Markov property makes the successive exit durations independent and identically distributed random variables that are strictly positive. Some fixed positive duration is exceeded with positive probability, so the Borel-Cantelli lemma makes the sum of the durations infinite. Translate and scale for other intervals. In particular, if a nonnegative function is bounded below by a positive constant on such an interval, its infinite-horizon Brownian occupation time integral diverges. Recurrent visits alone, without this duration argument, would not prove divergence of an integral.

The laws on continuous paths of a standard Brownian motion and a Brownian motion with drift $\mu \ne 0$ are mutually singular measures on the whole half-line. The event ${\lim }_{t\to \infty }\omega (t)/t=0$ has full measure for the former and zero measure for the latter by the strong law for Brownian motion. On every fixed finite horizon, their laws are equivalent by the Girsanov theorem. The finite-horizon densities $e^{\mu B_t-{\mu }^{2}t/2}$ tend to zero almost surely while retaining expectation one, so they lack uniform integrability over all time.

The natural Brownian filtration is the natural filtration generated by a Brownian motion. Its usual augmentation adds null events and makes it right-continuous; the Brownian martingale representation theorem holds for this augmented natural filtration. A larger Brownian filtration can preserve independence of future increments while containing additional randomness, so it need not have martingale representation with respect to the specified Brownian motion.

For standard Brownian motion, the Brownian bridge $U_t=B_t-t{B}_1$, $0\le t\le 1$, is independent of $B_1$. Every finite vector of bridge coordinates is jointly multivariate normal with $B_1$, and its cross-covariances with $B_1$ vanish. The independence of uncorrelated jointly normal variables and independence extended from generating pi-systems prove independence of the whole stochastic process.

For $p\in \partial J(v)$, zero Bregman distance means that $J(u)$ equals the supporting affine function $J(v)+⟨p,u-v⟩$. If $J$ is strictly convex, two distinct contact points are impossible: convexity along their segment would be strict, while the subgradient inequality supplies the opposite bound. Thus zero distance then implies $u=v$. Without strict convexity, $J(s)=|s|$, $v=1$, $p=1$ gives zero distance for every $u\ge 0$. These are all on the same supporting face.

A gradient-quartic Brazovskii model stabilizes a negative quadratic gradient coefficient with $\gamma ({\nabla }^2\varphi {)}^2/2$ and uses the quartic interaction $B{\varphi }^2|\nabla \varphi {|}^2$. For $\gamma ,B>0$ and a zero-average compositional order parameter, its nonzero-wavevector instability is a candidate for smectic phase ordering. The Gaussian variational kernel for a gradient quartic interaction shifts both the quadratic coefficients.

Over a splitting field for finite group representations of characteristic $p$, the irreducible Brauer characters form a complex basis of the class functions on the p-regular elements. Independence is the character form of the Brauer–Nesbitt theorem. To obtain spanning, extend such a class function by zero on the p-singular classes. Ordinary irreducible characters form a basis of all class functions by character orthogonality. Restricting them to p-regular elements yields Brauer characters of reductions of an integral form of a group representation in a compatible splitting p-modular system; if needed, first extend scalars, which does not change the simple-module list under the splitting hypothesis. Each restriction is a nonnegative integral sum of simple Brauer characters by exact-sequence additivity and the Jordan–Hölder theorem. Thus these restrictions span, proving the assertion. Consequently the number of simple modules equals the number of p-regular conjugacy classes, and evaluation identifies the complexified modular representation ring with the product of one copy of $\mathbb{C}$ for each such class.

A BPS state saturates a BPS bound in supersymmetry. The zero-norm combinations of supercharges and their adjoints annihilate its entire irreducible supermultiplet. Thus part of the supersymmetry is preserved. For a positive-mass representation, its massive supermultiplet is shortened; massless supermultiplets require the separate null-momentum construction. Charge conjugation can require a separate conjugate multiplet in the physical spectrum.

A bounded linear operator $T:X\to {\ell }_{\infty }(\Gamma )$ is exactly a uniformly bounded family of bounded linear functionals $f_{\gamma }\in X^{\ast }$ through $(Tx)(\gamma )=f_{\gamma }(x)$. Evaluation shows $\Vert f_{\gamma }\Vert \le \Vert T\Vert$; conversely the family bound gives $\Vert Tx\Vert \le (\sup \Vert f_{\gamma }\Vert )\Vert x\Vert$, proving equality. Taking all functionals in the dual unit ball gives an isometric embedding by the Hahn-Banach theorem.

The bounded piecewise-linear shear layer has exponentially growing modes exactly when $h<1/2$. Its squared phase speed tends to $2h-1$ as $kL\to 0$ and to $1$ as $kL\to \infty$. For $h<1/2$, sufficiently long nonzero waves are therefore unstable. Conversely $J-XY>0$, and for $h\ge 1/2$, $q\ge \alpha (1-h)>Y$, so $H-Y>0$ for every positive wavenumber and $c^2>0$. This proves the threshold without assuming monotonicity of the dispersion curve.

This construction adapts compactly supported orthonormal wavelets to an interval by finite changes near its endpoints. It preserves local support, polynomial reproduction, nested approximation spaces and orthogonal detail spaces. The resulting basis contains coarse scaling functions, unchanged interior wavelets, and finitely many boundary wavelets per endpoint and per scale, without forcing periodic or zero boundary values.